How to automatically/dynamically repel control points on a path/curve?
There is no built in tool to do this automatically, nor any specific tool to do what you want. I don't think blender as any sort of curve intersection tool, let alone one that takes bevel objects into account. 

That being said you could use a few other workarounds.

Either select all your intersecting vertex and scale them out relative to the selection center. That should move them apart eventually enough to avoid all intersections, or

You could use Proportional Editing for more localized control, to move only vertex within intersection radius. Select one vertex with close-by intersections, press O key to turn Proportional Editing on then scale it up. Control the influence radius with the Mousewheel to affect only neighbouring vertex in the desired influence area.